Let $(A^\bullet,\partial)$ be a complexe of $\mathbb{C}$-vector spaces. We suppose that this complex is of finite length, and all $A^\bullet$ are finite dimensional. Let $H^\bullet$ be the cohomology ...

Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...

I am interested in infinite loop structures on the infinite dimensional projective space $\mathbb{R} P^\infty$. Is it unique? I think this has to be known in work of May, and If so, then I presume its ...

I would like to know what structure has the category of liftings of a principal bundle. Let me be more precise.
Fix $k$ an algebraically closed field and $X$ a smooth projective variety over it (for ...

It is well-known that one can detect based loopspaces using the machinery of operads. Namely, given a group-like space $X$ with an action of $\mathbb{E}_n$-operad, then it is homotopy equivalent as an ...

What is the quotient space $\mathbb{T}^3/\mathbb{Z}_2$, When $\mathbb{Z}_2$ acting on 3-dimensional torus $\mathbb{T}^3 := \mathbb{R}^3/\mathbb{Z}^3$ by sending $x$ to $-x$? Does anyone know how to ...

One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between ...

Let $S_3$ be the permutation group of order $3$. Let $V_2(\mathbb{R}^n)$ be the stiefel manifold of $2$-frames in $\mathbb{R}^n$. Let $S_3$ act on $V_2(\mathbb{R}^n)$ by
$$
(1,2)(u,v)=(-u,v-u),
$$
$$
...

Let $h:\pi_{2n-1}(S^n) \rightarrow \mathbb{Z}$ be the Hopf invariant. I believe that in the same paper that proves his suspension theorem, Freudenthal proved that if $x \in \pi_{2n-1}(S^n)$ satisfies ...

It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).
...

Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$.
I find the following interpretation of $Z(G)$ in ...

As far as I know, when Poincare formulated his well known conjecture, the original statement was the follwoing: if a closed manifold has the same homology groups as the sphere it is homoeomorphic to ...

Suppose $X_1$ and $X_2$ are two spaces which has a common finite sheeted covering space $Z$ (may be distinct index)...from here can we conclude that there exists a space $W$ such that $X_1$ and $ X_2$ ...

Let $X$ be a connected pointed topological space equipped with two different actions of $E_\infty$-operad. Each action provides a collection of deloopings $X_i$, where $X_0 = X$ and $\Omega X_i$ is ...

I have been studying the BRST quantization in quantum field theory recently and noticed that the subject is very much related to algebraic topology and cohomology. A quick google search led me to the ...